By the definition of \(T_{z,v}\)  we must have \(T_{z,v} \le {\tau _{\text{exit}}}(z,v)\) .
Suppose that \(T_{z,v} < {\tau _{\text{exit}}}(z,v)\) . From Corollary REF  then we also know
\({\tau _{\text{cut}}}(z,v) = T_{z,v} < {\tau _{\text{exit}}}(z,v)\) 
Let \(p = \gamma _{z,v}(T_{z,v})\) , and by Lemma REF  it holds that \(p \in \overline{\sigma (z,v)} = \sigma (z,v)\) . By Lemma REF  there either exists a second distance minimizing geodesic from from \(z\)  to \(p\)  or \(p\)  is conjugate to \(z\)  along \(\gamma _{z,v}\) . Clearly in the first case \(r_p\)  is not differentiable at \(z\) . If the second case is valid, and since \(p\in M^{int}\) , we get by a similar proof as in [1]} that the exponential map \(\exp _{z}\)  is not a local injection at \(T_{z,v}v \in T_zM\) . This implies that there is a sequence of points \(p_i \in M^{int}\) , that converge to \(p\)  and can be connected to \(z\)  by at least two distance minimizing geodesics. Thus \(r_{p_i}\)  is not differentiable at \(z\)  for any \(i \in \mathbf {N}\)  implying that neither is \(r_p\) .
We arrive in a contradiction, meaning that the inequality (REF ) cannot occur and we must have \(T_{z,v} = {\tau _{\text{exit}}}(z,v)\) .
